 Research
 Open Access
 Published:
Results on exact controllability of secondorder semilinear control system in Hilbert spaces
Advances in Difference Equations volume 2021, Article number: 455 (2021)
Abstract
In our manuscript, we extend the controllability outcomes given by Bashirov (Math. Methods Appl. Sci. 44(9):7455–7462, 2021) for a family of secondorder semilinear control system by formulating a sequence of piecewise controls. This approach does not involve large estimations which are required to apply fixed point theorems. Therefore, we avoid the use of fixed point theory and the contraction mapping principle. We establish that a secondorder semilinear system drives any starting position to the required final position from the domain of the system. To achieve the required results, we suppose that the linear system is exactly controllable at every noninitial time period, the norm of the inverse of the controllability Grammian operator increases as the time approaches zero with the slower rate in comparison to the reciprocal of the square function, and the nonlinear term is bounded. Finally, an example has been presented to validate the results.
Introduction
Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time is known or postulated. This is illustrated in classical mechanics where the motion of a body is described by its position and velocity as time varies. For the studies related to the existence of solution for integer and fractionalorder systems, one can refer to [2–30]. The concept of controllability is one of the underlying ideas in mathematical control theory. Controllability analysis is used in several reallife problems which include, but are not limited to, rocket launching problems for satellite and aircraft control, missiles and antimissiles problems in defense, regulating inflation rate in the economy, controlling sugar level in the blood, etc. A systematic study of controllability was initiated by Kalman [31] in 1963 when the theory of controllability for timeinvariant and timevarying control systems in statespace form was developed.
Several engineering and scientific problems can be expressed by infinitedimensional differential equations. Therefore, it becomes necessary to discuss the controllability results for infinitedimensional systems. The controllability problems for finitedimensional nonlinear systems have been immensely analyzed by several authors. Many authors have advanced the idea of controllability from finitedimensional systems to infinitedimensional systems and determined appropriate requirements for the controllability of nonlinear systems. Different techniques have been practiced for the examination of controllability including fixed point theorems [32–34]. For more problems on controllability and recent progresses on fractional calculus and its applications, please refer to [35–47].
Secondorder differential equations represent abstract mathematical interpretations of several partial differential equations which occur in many applications related to the oscillation of fastened bars, the transverse motion of an extensible beam, and several other realworld physical phenomena. Hence it becomes really important to determine the controllability outcomes for this kind of system. Controllability discussions for various secondorder nonlinear systems have been widely studied by several authors [33, 48–58]. Fixed point theorems have been utilized greatly in determining the existence and controllability results for different firstorder and secondorder systems which involve large estimations on system constants see [33, 59, 60]. Recently Bashirov [1] obtained the exact controllability results for firstorder semilinear systems using a new technique that is based on the piecewise formulation of driving controls and without using fixed point theory. Earlier the same approach was applied in terms of approximate controllability [61]. Motivated by Bashirov [1], we extend the controllability results for secondorder semilinear systems excluding the use of fixed point theory. It is centered on a piecewise formulation of steering controls and does not involve large estimations which are required to apply fixed point theorems. By constructing a piecewise sequence of controls, we determine that a secondorder semilinear system is exactly controllable to the domain of the system operator, that is, it drives the system from any starting position to the required final position from the domain of the system. To achieve the required results, we suppose the subsequent requirements:

(a)
The corresponding linear system is exactly controllable at every noninitial time period.

(b)
The norm of the inverse of the controllability Grammian operator increases as the time approaches zero with the slower rate in comparison to the reciprocal of the square function.

(c)
The nonlinear term is bounded.
Let us consider \(Z=L_{2}[0,b;\mathbb{X}]\) and \(Y=L_{2}[0,b;\mathbb{U}]\) as the function spaces defined on \(J=[0,b]\), \(0\leq b<\infty \), where \(\mathbb{X}\) and \(\mathbb{U}\) are two Hilbert spaces. Consider the following secondorder semilinear control system:
where

1
\(p(t)\) represents the state having values in Hilbert space \(\mathbb{X}\).

2
Control function v is defined from \([0,b]\rightarrow \mathbb{U}\).

3
B is a bounded and linear operator from \(\mathbb{U}\) into \(\mathbb{X}\).

4
The function \(r:[0,b]\times \mathbb{X}\rightarrow \mathbb{X}\) is a purely nonlinear function which produces nonlinearity in the system.

5
\(A:\operatorname{dom}(A)\subseteq \mathbb{X}\rightarrow \mathbb{X}\) is linear, closed where \(\operatorname{dom}(A)\) is a dense subset of X.
The linear system corresponding to (1.1) with state vector \(q(t)\) and control v is defined by
The article is structured in the subsequent manner:

1
Sect. 2 presents a few basic results related to control theory and secondorder systems.

2
Sect. 3 provides the assumptions which are required to obtain the controllability results.

3
Sect. 4 discusses the controllability results using the new technique.

4
Sect. 5 presents an example to verify the established outcomes.
Auxiliary results
Here, we will review fundamental theories and a few definitions which would be helpful for further discussions.
Definition 2.1
[62] A oneparameter family \(\{C(t),t\in \mathbb{R}\}\) of bounded linear operators mapping the Hilbert space \(\mathbb{X}\) into itself is called a strongly continuous cosine family if and only if

1
\(C(0)=I\);

2
\(C(s+t)+C(st)=2C(s)C(t)\);

3
\(C(t)x\) is continuous in t on \(\mathbb{R}\) for each fixed \(x\in \mathbb{X}\).
If \(\{C(t),t\in \mathbb{R}\}\) is a strongly continuous cosine family in \(\mathbb{X}\), then \(\{S(t),t\in \mathbb{R}\}\) is a oneparameter family of operators in \(\mathbb{X}\) defined by
The infinitesimal generator of a strongly continuous cosine family \(\{C(t), t\in \mathbb{R}\}\) is the operator \(A:\mathbb{X}\rightarrow \mathbb{X}\) defined by
The domain of operator A is defined as
These cosine and sine families defined above and generator A fulfill the following properties.
Lemma 2.2
([28])
Suppose that A is the infinitesimal generator of a cosine family of operators \(\{C(t):t\in \mathbb{R}\}\). Then the following hold:

(1)
There exist \(M' \geq 1\) and \(\omega \geq 0\) such that \(\C(t)\\leq M'e^{\omega t}\), and hence \(\S(t)\\leq M'e^{\omega t}\).

(2)
\(A\int _{s}^{r}S(u)x\,du=[C(r)C(s)]x\) for all \(0\leq s\leq r<\infty \).

(3)
There exists \(N'\geq 1\) such that \(\S(s)S(r)\\leq N' \int _{s}^{r}e^{\omega s}\,ds \) for all \(0\leq s\leq r<\infty \).
The uniform boundedness principle together with \((1)\) implies that both \(\{C(t):t\in J\}\) and \(\{S(t):t\in J\}\) are uniformly bounded and \(M=M'e^{\omega b}\).
Proposition 2.3
([62])
Let \(\{C(t),t\in \mathbb{R}\}\) be a strongly continuous cosine family in \(\mathbb{X}\) with infinitesimal generator A. The following are true:

1
\(S(0)=0\).

2
\(C(t)=C(t)\) and \(S(t)=S(t)\) for all \(t\in \mathbb{R}\).

3
If \(x\in E\), then \(S(t)x,C(t)x\in \operatorname{dom}(A)\) and \(\frac{d}{dt}S(t)x=C(t)x\), and \(\frac{d}{dt}C(t)x=AS(t)x\), where \(E=\{x:C(t)x \textit{ is once continuously differentiable function of }t\}\).

4
If \(x\in \operatorname{dom}(A)\), then \(S(t)x\in \operatorname{dom}(A)\) and \(AS(t)x=S(t)Ax\).

5
If \(x\in \operatorname{dom}(A)\), then \(C(t)x\in \operatorname{dom}(A)\) and \(\frac{d^{2}}{dt^{2}}C(t)x=AC(t)x=C(t)Ax\).

6
If \(x\in E\), then \(\lim_{t\rightarrow 0}AS(t)x=0\).
Proposition 2.4
([62])
Let \(\{C(t), t \in \mathbb{R}\}\) be a strongly continuous cosine family in \(\mathbb{X}\). The operator \(\hat{A}: \mathbb{X}\rightarrow \mathbb{X}\) defined by
with domain \(x\in \mathbb{X}\) for which this limit exists, is the infinitesimal generator of the cosine family \(\{C(t), t\in \mathbb{R}\}\).
Suppose \(U_{ad}=L_{2}[0,b;\mathbb{U}]\), which is the set of admissible controls. We define the mild solution of the given semi linear system (1.1) and its corresponding linear system as follows.
Definition 2.5
The mild solution of system (1.1) is defined by a function \(p(\cdot )\in \mathbb{X}\) which satisfies the following integral equation:
and the mild solution of the corresponding linear system (1.2) is described by the following integral equation:
Definition 2.6
([1])
System (1.1) is said to be approximately controllable in the time interval \([0,b]\) if, for the given starting position \((p_{0}, q_{0})\in \mathbb{X}\) and the required final position \((p_{F}, q_{F})\in \mathbb{X}\) and \(\epsilon >0\), there exists a control function \(v\in U_{ad}\) such that the solution of (1.1) satisfies
where \(p(b)\) is the state value of system (1.1) at time \(t=b\). If \(p(b)=p_{F}\), then the system is said to be exactly controllable. The system is said to be exactly controllable to \(\operatorname{dom}(A)\) on \([0,b]\) if, for the given starting position \((p_{0}, q_{0})\in \mathbb{X}\) and the required final position \((p_{F}, q_{F})\in \operatorname{dom}(A)\), there exists a control function \(v\in U_{ad}\) such that the solution of the system satisfies \(p(b)=p_{F}\), \(p'(b)=q_{F}\).
Remark 2.7
Note that the exact controllability to \(\operatorname{dom}(A)\) lies in between the exact and approximate controllability. Therefore, it is a weaker concept than the exact controllability. In real life applications, sometimes we are more concerned with attaining the points from \(\operatorname{dom}(A)\). If it is possible to reach the points from \(\mathbb{X} \backslash \operatorname{dom}(A)\) as well, then it can be considered as an additional capability of the system.
Assumptions
Let us introduce the controllability Grammian operator W associated with linear system (1.2) by
where \(S^{*}(s)\) denotes the adjoint of \(S(s)\).
Theorem 3.1
The corresponding linear system (1.2) is exactly controllable on the interval \([h, b]\) iff \(W(bh)\) is coercive. The control which drives the system from the starting position \((q(h), q'(h))\in X\) to the final position \((p_{F}, q_{F})\in X\) is given by
Proof
The result can be seen in [59]. Moreover, it can be easily verified by substituting the above defined control in the mild solution of the corresponding linear system that it transfers \((q(h), q'(h))\) to \((p_{F}, q_{F})\) on the interval \([h, b]\). □
Remark 3.2
The coercivity of \(W(t)\) indicates that \((W(t))^{1}\) is a bounded linear operator. We say that \(W(t)\) is coercive if there exists \(\gamma >0\) such that \(\langle W(t)x, x\rangle \geq \gamma \x\^{2}\) for all \(x\in X\). Here \(W(0)=0\) and, therefore, it fails to be coercive. But it may be coercive for \(0< t\leq b\). Therefore, the above result holds on \([h, b]\).
To determine the main result, we make the following assumptions on the controllability Grammian operator W and the nonlinear function \(r(t,p)\):

(I)
\(W(t)\) is coercive for all \(0< t\leq b\).

(II)
There exists some \(N\geq 0\) such that
$$ t^{1+\alpha } \bigl\Vert \bigl(W(t)\bigr)^{1} \bigr\Vert \leq N \quad \text{for all } 0< t\leq b, 0\leq \alpha < 1. $$That is, \(\(W(t))^{1}\ \rightarrow \infty \) as t → 0^{+} with the slower rate in comparison to the reciprocal of the square function as
$$ \bigl\Vert \bigl(W(t)\bigr)^{1} \bigr\Vert \leq \frac{N}{t^{1+\alpha }}< \frac{N}{t^{2}}, $$for small values of t.

(III)
The nonlinear function r is Lebesgue measurable in t.

(IV)
r is Lipschitz continuous in p.

(V)
r is bounded in \([0,b]\times \mathbb{X}\), \(i.e\)., there exists \(K>0\) such that
$$ \bigl\Vert r(t,p) \bigr\Vert \leq K \quad \text{for all } (t,p)\in [0,b] \times \mathbb{X}. $$
Results on controllability
In this section, we primarily focus on the study of exact controllability of the assumed system.
Theorem 4.1
System (1.1) is exactly controllable to \(\operatorname{dom}(A)\) on the interval \([0,b]\) for every \(b>0\) provided assumptions (I)–(V) hold.
Proof: We construct a piecewise sequence of driving controls to formulate the required control function v which drives the given system from the starting position \((p_{0}, q_{0})\in \mathbb{X}\) to the final position \((p_{F}, q_{F})\in \operatorname{dom}(A)\) in the following manner.
For this, consider the sequence \(\{h_{n}\}\) which is defined by \(h_{n}=\frac{b}{2^{n}}\) for \(n=1,2,\ldots \) .
We have \(\sum_{n=1}^{\infty }h_{n}=b\). For the sake of simplicity, let us take \(h_{0}=0\) and
Then \(\lim_{n\rightarrow \infty }b_{n}=\sum_{k=0}^{\infty }h_{k}=b\).
Using Theorem 3.1, the corresponding linear system (1.2) is exactly controllable on \([b_{0}, b_{1}]\) along with the control
which steers the initial state \(p_{0}\) to \(C(h_{1})p_{F}+S(h_{1})q_{F}\).
That is,
Define v on \([b_{0},b_{1}]\) by letting \(v(\varrho )=v_{1}(\varrho )\). Then, from (2.1), we obtain
For brevity, let \(p(b_{1})=p_{1}\). Next, consider (1.2) on \([b_{1},b_{2}]\). By Theorem 3.1, the control
steers \(p_{1}\) to \(C(h_{2})p_{F}+S(h_{2}) q_{F}\). Writing \(p'(b_{1})=q_{1}\), then the control \(u_{2}(\varrho )\) can be written as
That is,
Define v on \((b_{1},b_{2}]\) by letting \(v(\varrho )=v_{2}(\varrho )\). Then, from (2.1), we obtain
For the sake of convenience, let \(p(b_{2})=p_{2}\). Progressing in this fashion, we acquire a sequence of driving controls
where \(q_{n1}=p'(b_{n1})\).
After combining the above sequence of controls, we get the control function as follows:
and
Now by using the assumption (V), we get
where \(M=\sup_{[0,b]}\C(\varrho )\\) and \(K=\sup_{[0,b]\times X}\r(\varrho ,p)\\).
Since \(C(\varrho )\) is strongly continuous, \(S(0)=0\) and \(\lim_{n\rightarrow \infty }h_{n}=0\).
Therefore, \(\lim_{n\rightarrow \infty }p_{n}=p_{F}\). Also,
Thus, we have
Since \(C(\varrho )\) is strongly continuous, \(S(0)=0\) and \(\lim_{n\rightarrow \infty }h_{n}=0\).
Therefore, \(\lim_{n\rightarrow \infty }q_{n}=q_{F}\).
Next we prove that \(v\in U_{ad}\). Since every \(v_{n}\) of v is continuous on the interval \((b_{n1},b_{n}]\) for \(n=0,1,2,\ldots \) , hence v is measurable. Also,
Therefore, by (4.3) and (4.4),
Since \(C(h_{0})p_{F}=p_{F}\) and \(KMh_{0}=0\), we obtain
Since \(\lim_{\varrho \rightarrow 0} \frac{C(2t)p_{F}p_{F}}{2t^{2}}=\hat{A}p_{F}\), using proposition (2) results in
which implies
for some \(P, P'>0\).
Then, by using assumption (II),
where \(\q_{F}\\leq C\) and \(\A p_{F} \\leq L\) for some \(C, L>0\).
Since \(0<\alpha <1\), therefore the series
\(\sum_{n=1}^{\infty } \frac{h_{n}}{(h_{n+1})^{\alpha 1}}\) and \(\sum_{n=1}^{\infty } \frac{h_{n}^{2}}{(h_{n+1})^{\alpha 1}}\) are convergent.
This shows that \(v\in U_{ad}\). Therefore, p is continuous and
Thus, \(p(b)=p_{F}\) and \(p'(b)=q_{F}\).
Example
Consider the partial differential equation:
Let \(\mathbb{X}=L_{2}[0,\pi ]\) and \(\gamma :[0,b]\times (0,\pi )\rightarrow \mathbb{R}\) be a continuous control function in ϱ. Define the operator \(A:D(A)\rightarrow \mathbb{X}\) by
with \(D(A)=\{\eta \in \mathbb{X}:\eta ,\eta ^{\prime } \text{ are absolutely continuous } \eta ^{\prime \prime }\in \mathbb{X},\eta (0)=\eta (\pi )=0\} \). A is an infinitesimal generator of a strongly continuous cosine family \(C(\varrho )\) on X. Moreover, the spectrum of A consists of eigenvalues \(n^{2}\) for \(n=1,2,3,\ldots \) , with the associated normalized eigenvectors \(\eta _{n}(s)=(2/\pi )^{1/2}\sin (ns)\). In particular,
The cosine function \(C(\varrho )\) and the sine function \(S(\varrho )\) are defined in the following way:
respectively. Define \(r:J \times \mathbb{X}\rightarrow \mathbb{X}\) by
Let \(v:[0,b]\rightarrow \mathbb{U}\) be defined by
Define the controllability operator in the following way:
Let us choose the system operator A in such a way that \(W(\varrho )\) is coercive for all \(0< \varrho \leq b\) and there exists some \(N\geq 0\) such that
Also, the nonlinear function σ can be considered satisfying conditions (III)–(V).
The considered PDE (5.1) can be converted to (1.1). Therefore, system (5.1) is exactly controllable to \(\operatorname{dom}(A)\).
Conclusion
In the present manuscript, the exact controllability to \(\operatorname{dom}(A)\) for a secondorder semilinear system has been discussed using a new technique which avoids fixed point theorems and does not involve large estimations on the system constants. The control function has been formulated by the piecewise construction of steering controls. These results can be further extended for systems with delay or deviated arguments with impulses and fractionalorder systems.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Bashirov, A.E.: On exact controllability of semilinear systems. Math. Methods Appl. Sci. 44(9), 7455–7462 (2021). https://doi.org/10.1002/mma.6265
Afshari, H., Shojaat, H., Moradi, M.S.: Existence of the positive solutions for a tripled system of fractional differential equations via integral boundary conditions. Results Nonlinear Anal. 4(3), 186–193 (2021). https://doi.org/10.53006/rna.938851
Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional differential equations via coupled fixed point. Electron. J. Differ. Equ. 15, 286 (2015) http://ejde.math.txstate.edu
Afshari, H., Gholamyan, H., Zhai, C.B.: New applications of concave operators to existence and uniqueness of solutions for fractional differential equations. Math. Commun. 25(1), 157–169 (2020)
Abdeljawad, T., Agarwal, R.P., Karapinar, E., Kumari, P.S.: Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended bmetric space. Symmetry 11(5), 686 (2019). https://doi.org/10.3390/sym11050686
Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6652
Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solutions of fractional differential equations via Geraghty type hybrid contractions. Appl. Comput. Math. 20(2), 313–333 (2021)
Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: Uniqueness of solution for higherorder nonlinear fractional differential equations with multipoint and integral boundary conditions. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 2021, 155 (2021). https://doi.org/10.1007/s13398021010953
Sabetghadam, F., Masiha, H.P.: Fixedpoint results for multivalued operators in quasiordered metric spaces. Appl. Math. Lett. 25(11), 1856–1861 (2012). https://doi.org/10.1016/j.aml.2012.02.046
Masiha, H.P., Sabetghadam, F., Shahzad, N.: Fixed point theorems in partial metric spaces with an application. Filomat 27(4), 617–624 (2013)
Sabetghadam, F., Masiha, H.P., Altun, I.: Fixedpoint theorems for integraltype contractions on partial metric spaces. Ukr. Math. J. 68, 940–949 (2016). https://doi.org/10.1007/s1125301612675
Baleanu, D., Jajarmi, A., Mohammadi, H., Rezapour, S.: A new study on the mathematical modelling of human liver with CaputoFabrizio fractional derivative. Chaos Solitons Fractals 134, 109705 (2020). https://doi.org/10.1016/j.chaos.2020.109705
Baleanu, D., Mohammadi, H., Rezapour, S.: Analysis of the model of HIV1 infection of \(CD4^{+}\) Tcell with a new approach of fractional derivative. Adv. Differ. Equ. 2020, 71 (2020). https://doi.org/10.1186/s1366202002544w
Mohammadi, H., Kumar, S., Rezapour, S., Etemad, S.: A theoretical study of the CaputoFabrizio fractional modeling for hearing loss due to Mumps virus with optimal control. Chaos Solitons Fractals 144, 110668 (2021). https://doi.org/10.1016/j.chaos.2021.110668
Rezapour, S., Mohammadi, H., Jajarmi, A.: A new mathematical model for Zika virus transmission. Adv. Differ. Equ. 2020, 589 (2020). https://doi.org/10.1186/s13662020030447
Rezapour, S., Henriquez, H.R., Vijayakumar, V., Nisar, K.S., Shukla, A.: A note on existence of mild solutions for secondorder neutral integrodifferential evolution equations with statedependent delay. Fractal Fract. 5(3), 126 (2021). https://doi.org/10.3390/fractalfract5030126
Rezapour, S., Imran, A., Hussain, A., Martinez, F., Etemad, S., Kaabar, M.K.A.: Condensing functions and approximate endpoint criterion for the existence analysis of quantum integrodifference FBVPs. Symmetry 13(3), 469 (2021). https://doi.org/10.3390/sym13030469
Rezapour, S., Azzaoui, B., Tellab, B., Etemad, S., Masiha, H.P.: An analysis on the positivesSolutions for a fractional configuration of the Caputo multiterm semilinear differential equation. J. Funct. Spaces 2021, Article ID 6022941 (2021). https://doi.org/10.1155/2021/6022941
Baleanu, D., Etemad, S., Rezapour, S.: On a fractional hybrid integrodifferential equation with mixed hybrid integral boundary value conditions by using three operators. Alex. Eng. J. 59(5), 3019–3027 (2020). https://doi.org/10.1016/j.aej.2020.04.053
Panda, S.K., Karapinar, E., Atangana, A.: A numerical schemes and comparisons for fixed point results with applications to solutions of Volterra integral equations in dislocated extended bmetric space. Alex. Eng. J. 59(2), 815–827 (2020). https://doi.org/10.1016/j.aej.2020.02.007
Hazarika, B., Karapinar, E., Arab, R., Rabbani, M.: Metriclike spaces to prove existence of solution for nonlinear quadratic integral equation and numerical method to solve it. J. Comput. Appl. Math. 328, 302–313 (2018). https://doi.org/10.1016/j.cam.2017.07.012
Aydogan, S.M., Baleanu, D., Mousalou, A., Rezapour, S.: On high order fractional integrodifferential equations including the CaputoFabrizio derivative. Bound. Value Probl. 2018, 90 (2018). https://doi.org/10.1186/s1366101810089
Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some infinite coefficientsymmetric CaputoFabrizio fractional integrodifferential equations. Bound. Value Probl. 2017, 145 (2017). https://doi.org/10.1186/s1366101708679
Baleanu, D., Rezapour, S., Saberpour, Z.: On fractional integrodifferential inclusions via the extended fractional CaputoFabrizio derivation. Bound. Value Probl. 2019, 79 (2019). https://doi.org/10.1186/s1366101911940
Samei, M.E., Rezapour, S.: On a system of fractional qdifferential inclusions via sum of two multiterm functions on a time scale. Bound. Value Probl. 2020, 135 (2020). https://doi.org/10.1186/s13661020014331
Rezapour, S., Samei, M.E.: On the existence of solutions for a multisingular pointwise defined fractional qintegrodifferential equation. Bound. Value Probl. 2020, 38 (2020). https://doi.org/10.1186/s13661020013423
Baleanu, D., Etemad, S., Rezapour, S.: A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions. Bound. Value Probl. 2020, 64 (2020). https://doi.org/10.1186/s13661020013610
Fattorini, H.O.: Second Order Linear Differential Equations in Banach Spaces. NorthHolland, Amsterdam (1995)
Neshati, J., Masiha, H.P., Jafarian, M.: Electrochemical noise analysis for estimation of corrosion rate of carbon steel in crude oil. AntiCorros. Methods Mater. 54(1), 27–33 (2007). https://doi.org/10.1108/00035590710717366
Neshati, J., Masiha, H.P., Mahjani, M.G., Jafarian, M.: Electrochemical noise analysis for estimation of corrosion rate of carbon steel in crude oil. Corros. Eng. Sci. Technol. 42(4), 371–376 (2007). https://doi.org/10.1179/174327807X214879
Kalman, R.: Lectures on Controllability and Observability. Springer, Berlin (2010). https://doi.org/10.1007/9783642110634_1
Arthi, G., Balachandran, K.: Controllability of secondorder impulsive evolution systems with infinite delay. Nonlinear Anal. Hybrid Syst. 11, 139–153 (2014). https://doi.org/10.1016/j.nahs.2013.08.001
Sukavanam, S., Sukavanam, N.: Controllability of secondorder systems with nonlocal conditions in Banach spaces. Numer. Funct. Anal. Optim. 35(4), 423–431 (2014). https://doi.org/10.1080/01630563.2013.814067
Leiva, H.: Rothe’s fixed point theorem and controllability of semilinear nonautonomous systems. Syst. Control Lett. 67, 14–18 (2014). https://doi.org/10.1016/j.sysconle.2014.01.008
Bashirov, A.E., Mahmudov, N.I.: On concepts of controllability for deterministic and stochastic systems. SIAM J. Control Optim. 37(6), 1808–1821 (1999). https://doi.org/10.1137/S036301299732184X
Chalishajar, D.N.: Controllability of mixed VolterraFredholmtype integrodifferential systems in Banach space. J. Franklin Inst. 344(1), 12–21 (2007). https://doi.org/10.1016/j.jfranklin.2006.04.002
Kavitha, K., Vijayakumar, V., Udhayakumar, R.: Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness. Chaos Solitons Fractals 139, 110035 (2020). https://doi.org/10.1016/j.chaos.2020.110035
Klamka, J.: Controllability of dynamical systems, a survey. Bull. Pol. Acad. Sci., Tech. Sci. 61(2), 335–342 (2013). https://doi.org/10.2478/bpasts20130031
Klamka, J., Babiarz, A., Niezbitowski, M.: Banach fixedpoint theorem in semilinear controllability problems, a survey. Bull. Pol. Acad. Sci., Tech. Sci. 64(1), 21–35 (2016). https://doi.org/10.1515/bpasts20160004
Klamka, J., Wyrwal, J., Zawiski, R.: On controllability of second order dynamical systems, a survey. Bull. Pol. Acad. Sci., Tech. Sci. 65(3), 279–295 (2017). https://doi.org/10.1515/bpasts20170032
Naito, K.: Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. 25(3), 715–722 (1987). https://doi.org/10.1137/0325040
Raja, M.M., Vijayakumar, V., Udhayakumar, R., Zhou, Y.: A new approach on the approximate controllability of fractional differential evolution equations of order \(1 < r < 2\) in Hilbert spaces. Chaos Solitons Fractals 141, 110310 (2020). https://doi.org/10.1016/j.chaos.2020.110310
Shukla, A., Patel, R.: Controllability results for fractional semilinear delay control systems. J. Appl. Math. Comput. 65, 861–875 (2021). https://doi.org/10.1007/s12190020014184
Vijayakumar, V., Udhayakumar, R., Kavitha, K.: On the approximate controllability of neutral integrodifferential inclusions of Sobolevtype with infinite delay. Evol. Equ. Control Theory 10(2), 271–296 (2021). https://doi.org/10.3934/eect.2020066
Kumar, V., Malik, M.: Total controllability and observability for dynamic systems with noninstantaneous impulses on time scales. Asian J. Control 23(2), 847–859 (2021). https://doi.org/10.1002/asjc.2268
Vijayakumar, V., Murugesu, R., Poongodi, R., Dhanalakshmi, S.: Controllability of second order impulsive nonlocal Cauchy problem via measure of noncompactness. Mediterr. J. Math. 14(1), 1–23 (2017). https://doi.org/10.1007/s0000901608136
Vijayakumar, V., Murugesu, R., Selvan, M.T.: Controllability for a class of second order functional evolution differential equations without uniqueness. IMA J. Math. Control Inf. 36(1), 225–246 (2019). https://doi.org/10.1093/imamci/dnx048
Balachandran, K., Anthoni, S.M.: Controllability of secondorder semilinear neutral functional differential systems in Banach spaces. Comput. Math. Appl. 41(10–11), 1223–1235 (2001). https://doi.org/10.1016/S08981221(01)000931
Haq, A., Sukavanam, N.: Controllability of secondorder nonlocal retarded semilinear systems with delay in control. Appl. Anal. 99(16), 2741–2754 (2020). https://doi.org/10.1080/00036811.2019.1582031
Henriquez, H.R., Hernandez, E.: Approximate controllability of secondorder distributed implicit functional systems. Nonlinear Anal., Theory Methods Appl. 70(2), 1023–1039 (2009). https://doi.org/10.1016/j.na.2008.01.029
Sukavanam, S., Tomar, N.K.: Mild solution and controllability of secondorder nonlocal retarded semilinear systems. IMA J. Math. Control Inf. 37(1), 39–49 (2020). https://doi.org/10.1093/imamci/dny037
Mahmudov, N.I., Udhayakumar, R., Vijayakumar, V.: On the approximate controllability of secondorder evolution hemivariational inequalities. Results Math. 75, 160 (2020). https://doi.org/10.1007/s00025020012932
Mahmudov, N.I., Vijayakumar, V., Murugesu, R.: Approximate controllability of secondorder evolution differential inclusions in Hilbert spaces. Mediterr. J. Math. 13(5), 3433–3454 (2016). https://doi.org/10.1007/s0000901606957
Sakthivel, R., Mahmudov, N.I., Kim, J.H.: On controllability of second order nonlinear impulsive differential systems. Nonlinear Anal., Theory Methods Appl. 71(1–2), 44–52 (2009). https://doi.org/10.1016/j.na.2008.10.029
Shukla, A., Sukavanam, N., Pandey, D.N., Arora, U.: Approximate controllability of secondorder semilinear control system. Circuits Syst. Signal Process. 35, 3339–3354 (2016). https://doi.org/10.1007/s0003401501915
Shukla, A., Patel, R.: Existence and optimal control results for secondorder semilinear system in Hilbert spaces. Circuits Syst. Signal Process. 40, 4246–4258 (2021). https://doi.org/10.1007/s00034021016802
Vijayakumar, V., Murugesu, R.: Controllability for a class of secondorder evolution differential inclusions without compactness. Appl. Anal. 98(7), 1367–1385 (2019). https://doi.org/10.1080/00036811.2017.1422727
Vijayakumar, V., Udhayakumar, R., Dineshkumar, C.: Approximate controllability of second order nonlocal neutral differential evolution inclusions. IMA J. Math. Control Inf. 38(1), 192–210 (2021). https://doi.org/10.1093/imamci/dnaa001
Curtain, R.F., Zwart, H.J.: An Introduction to Infinite Dimensional Linear Systems Theory. Springer, New York (1995)
Zhou, H.X.: Approximate controllability for a class of semilinear abstract equations. Asian J. Control 21(4), 551–565 (1983). https://doi.org/10.1137/0321033
Bashirov, A.E., Ghahramanlou, N., Lam, J.: On partial approximate controllability of semilinear systems. Cogent Eng. 1(1), 965947 (2014). https://doi.org/10.1080/23311916.2014.965947
Travis, C.C., Webb, G.F.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hung. 32, 75–96 (1978). https://doi.org/10.1007/BF01902205
Acknowledgements
The fifth author was supported by Azarbaijan Shahid Madani University. The authors express their gratitude to dear unknown referees for their helpful suggestions which improved the final version of this paper.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.
Corresponding authors
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Arora, U., Vijayakumar, V., Shukla, A. et al. Results on exact controllability of secondorder semilinear control system in Hilbert spaces. Adv Differ Equ 2021, 455 (2021). https://doi.org/10.1186/s13662021036205
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662021036205
MSC
 34K30
 34K35
 93C25
Keywords
 Mild solution
 Controllability
 Secondorder system
 Cosine family